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Classification of Thurston-relation subfactor planar algebras

Corey Jones Ohio State University, Columbus, USA Zhengwei Liu Harvard University, Cambridge, USA Yunxiang Ren Harvard University, Cambridge, USA

Quantum Algebra Spectral Theory and Operator Algebra mathscidoc:2207.29002

European Mathematical Society, 10, (3), 441-480, 2019.9
Bisch and Jones proposed the classification of planar algebras by simple generators and relations. They investigated with the second author the classification of planar algebras generated by 2-boxes. In this paper, we classify singly-generated Thurston-relation planar algebras, defined as subfactor planar algebras generated by a 3-box satisfying a relation proposed by Dylan Thurston. Our main result shows that such subfactor planar algebras are either the E6 subfactor planar algebras or belong to a two-parameter family of planar algebras arising from the representations of type A quantum groups. We introduce a new method for determining positivity of the Markov trace of planar algebras in this family.
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@inproceedings{corey2019classification,
  title={Classification of Thurston-relation subfactor planar algebras},
  author={Corey Jones, Zhengwei Liu, and Yunxiang Ren},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220703160517704973535},
  booktitle={European Mathematical Society},
  volume={10},
  number={3},
  pages={441-480},
  year={2019},
}
Corey Jones, Zhengwei Liu, and Yunxiang Ren. Classification of Thurston-relation subfactor planar algebras. 2019. Vol. 10. In European Mathematical Society. pp.441-480. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220703160517704973535.
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